# Solving Rational Equations

A rational equation is an equation with rational expressions on either side of the equals sign.

ONE TECHNIQUE for solving rational equations is cross-multiplication — what some textbooks call the means/extremes property .

This method works only if on each side of the equation there is only one rational expression.

Example 1:

Solve:

$\frac{7}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}=\frac{x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}$

Cross multiplying, we get:

${x}^{2}+2x=7x+14$

This quadratic equation can be solved by factoring .

${x}^{2}-5x-14=0$

$\left(x-7\right)\left(x+2\right)=0$

Remember to check in the original equation for validity of solutions. In this case, $x=7$ is valid but $x=-2$ isn't, since it means division by zero in the original equation.

ANOTHER METHOD is to multiply through by the least common denominator of all of the fractions on either side of the equation.

Example 2:

Solve:

$\frac{x}{16}-\frac{3}{8x}=\frac{5}{16}$

The least common denominator (LCD) in this case is $16x$ . So, multiply both sides of the equation by $16x$ .

$\frac{x\left(16x\right)}{16}-\frac{3\left(16x\right)}{8x}=\frac{5\left(16x\right)}{16}$

${x}^{2}-6=5x$

Solve the quadratic equation by factoring.

${x}^{2}-5x-6=0$

$\left(x-6\right)\left(x+1\right)=0$

$x=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=-1$

Remember to check back to make sure these solutions are valid – that is, that they don't result in division by zero when substituted in the original equation. In this case, both solutions are valid.